Logarithmic Convergence

Logarithmic Convergence is the name the book gives to the steady approach of successive harmonic ratios toward 1 as recursion depth increases, which compresses the spectrum. With pulse density P(n) = (n+1)², the ratio of each term to the one before starts large and narrows: R(0) = 4.000, R(1) = 2.250, R(2) ≈ 1.778, R(3) = 1.563, R(4) = 1.440, tending to 1 as n grows. The book reads this as systematic spectral compression in nonlinear resonance systems, distinct from classical uniform harmonic distributions, and as the source of Spectral Clustering. Among the Chapter 7.2 predictions, this ratio convergence should appear in nonlinear acoustic systems through successive harmonic measurements. The formal statements are the Successive Term Ratio R(n) = P(n+1)/P(n) = (n + 2)²/(n + 1)², its limit lim(n→∞) R(n) = 1, and the Compression Rate Function dR/dn = -2(n+2)/(n+1)³, where R(n) is the successive term ratio, P(n) the pulse density function and n the recursion depth index. All are dimensionless [∅]; the book's checks are valid. It connects to the Successive Term Ratio, Spectral Clustering, Harmonic Origin Pulse, Pulse Density and Harmonic Clustering Coefficient.